Passive Scalar Interactions

Volume-Weighted Interaction

Passive scalars have no effect on the physical activity in the simulation, but can be thought of as massless tracer dyes. Passive scalar interactions between Lagrangian particles and the Eulerian fluid allow the particles to absorb passive scalars from the fluid or diffuse passive scalars into it. In this way, levels of passive scalars can measure effects such as residence times or degrees of exposure, or track particles through a mesh of cells.

Volume-weighted interactions model diffusion from the Lagrangian particle into the Eulerian fluid. This interaction conserves the density-weighted passive scalar.

The volume-weighted interaction models the mass flow rate as:

Figure 1. EQUATION_DISPLAY

J = k (ρ_{1} φ_{1} - ρ_{2} φ_{2}) A_{p}

(3318)

where:

$J$ is the mass flow rate.
$k$ is the interaction coefficient set under the Phase Interaction node.
$ρ_{1}$ is the density of the Lagrangian particle.
$ρ_{2}$ is the density of the Eulerian fluid.
$φ_{1}$ is the value of the passive scalar for the Lagrangian particle.
$φ_{2}$ is the value of the passive scalar for the Eulerian fluid.
$A_{p}$ is the surface area of the particle.

The source term for a Lagrangian particle in a parcel describes the rate of change of the passive scalar on the particle. A typical term is:

Figure 2. EQUATION_DISPLAY

S_{φ_{1}} = (- J) / V_{p}

(3319)

where:

$S_{φ_{1}}$ is the value of the passive scalar source for the particle.
$V_{p}$ is the volume of the particle.

The source term for the Eulerian cell is:

Figure 3. EQUATION_DISPLAY

S_{φ_{2}} = n_{p} \cdot J / V_{c}

(3320)

where:

$n_{p}$ is the number of particles in the parcel.
$S_{φ_{2}}$ is the value of the passive scalar source for the fluid.
$V_{c}$ is the volume of the cell.

Area-Weighted Interaction

Area-weighted interactions model adsorption/desorption between the Lagrangian particle and the Eulerian fluid, with distinct forward and backward rates.

The area-weighted interaction models the mass flow rate as:

Figure 4. EQUATION_DISPLAY

J = (k_{1 \to 2} ρ_{1} φ_{1} - k_{2 \to 1} ρ_{2} φ_{2}) A_{p}

(3321)

where:

$J$ is the mass flow rate.
$k_{1 \to 2}$ and $k_{2 \to 1}$ are the interaction coefficients set under the Phase Interaction node.
$ρ_{1}$ is the density of the Lagrangian particle.
$ρ_{2}$ is the density of the Eulerian fluid.
$φ_{1}$ is the value of the passive scalar for the Lagrangian particle.
$φ_{2}$ is the value of the passive scalar for the Eulerian fluid.
$A_{p}$ is the surface area of the particle.

The source term for a typical Lagrangian particle in the parcel is:

Figure 5. EQUATION_DISPLAY

S_{φ_{1}} = (- J) / A_{p}

(3322)

where:

$S_{φ_{1}}$ is the value of the passive scalar source for the particle.
$A_{p}$ is the surface area of the particle.

The source term for the Eulerian cell is:

Figure 6. EQUATION_DISPLAY

S_{φ_{2}} = n_{p} \cdot J / V_{c}

(3323)

where:

$n_{p}$ is the number of particles in the parcel.
$S_{φ_{2}}$ is the value of the passive scalar source for the fluid.
$V_{c}$ is the volume of the cell.

Passive Scalar Transfer for DEM Particles

The Passive Scalar Transfer model describes how a passive scalar rubs off one DEM particle onto another. It is primarily intended for describing how a coating of negligible mass is shared among colliding particles.

The area-weighted and volume-weighted transfer methods are both described by this equation:

\dot{m} = k (C_{1} ρ_{1} n_{1} - C_{2} ρ_{2} n_{2}) A_{c}

(3324)

where:

$\dot{m}$ is the flow of passive scalar between phases 1 and 2, regarded as analogous to a mass flux.
$k$ is the value of the user-set property Coefficient in the Area Weighted or Volume Weighted node.
$C_{1}$ and $C_{2}$ are the values of the passive scalar on phase 1 and phase 2.
$ρ_{1}$ and $ρ_{2}$ are the particle densities of phase 1 and phase 2.
$n_{1}$ and $n_{2}$ are the parcel particle counts for phase 1 and phase 2.
$A_{c}$ is the area that is computed from the harmonic mean of the radii of the two particles.

In volume-weighted transfer, the flow is divided by the volume of the respective particles, while in area-weighted transfer, the flow is divided by the surface areas.

In the Increment transfer method, a fixed amount of passive scalar (the increment, which is given by Coefficient) is added to each particle every time they make contact. This method does not model transfer of any physical property, but can be used in cases such as counting the number of contacts a particle undergoes.

In the Source transfer method, the amount of passive scalar transferred is integrated over time for the duration of the contact, for both particles:

C_{s} (t + d t) = C_{s} (t) + k ρ_{p} d t

(3325)

where:

$C_{s} (t)$ is the value of the passive scalar at time $t$ .
$k$ is the interaction coefficient.
$ρ_{p}$ is the particle density.